scholarly journals Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

2015 ◽  
pp. pdv047 ◽  
Author(s):  
Nicola Gigli ◽  
Andrea Mondino ◽  
Giuseppe Savaré
Keyword(s):  
Ricci Curvature ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Heat Flows ◽  
Measure Spaces

scholarly journals Lagrangian calculus for nonsymmetric diffusion operators

2020 ◽  
Vol 13 (4) ◽  
pp. 361-383
Author(s):  
Christian Ketterer
Keyword(s):  
Ricci Curvature ◽  
Vector Fields ◽  
Probabilistic Approach ◽  
Gradient Estimates ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Maximal Diameter ◽  
Measure Spaces ◽  
Evolution Variational Inequality ◽  
Dual Semigroup

AbstractWe characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the {L^{2}}-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N-warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

scholarly journals Contraction and regularizing properties of heat flows in metric measure spaces

2021 ◽  
Vol 14 (1) ◽  
pp. 273-297
Author(s):  
Giulia Luise ◽  
◽  
Giuseppe Savaré ◽  
Keyword(s):  
Metric Measure Spaces ◽  
Heat Flows ◽  
Measure Spaces

scholarly journals Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

2020 ◽  
Vol 30 (6) ◽  
pp. 1648-1711
Author(s):  
Karl-Theodor Sturm
Keyword(s):  
Lower Bound ◽  
Second Fundamental Form ◽  
Gradient Estimates ◽  
List Type ◽  
Metric Measure Spaces ◽  
Heat Flows ◽  
Measure Spaces ◽  
Neumann Laplacian ◽  
Boundary Curvature ◽  
Time Changes

AbstractWe will study metric measure spaces $$(X,\mathsf{d},{\mathfrak {m}})$$ ( X , d , m ) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds $$\mathsf{BE}_1(\kappa ,\infty )$$ BE 1 ( κ , ∞ ) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary $$\psi \in \mathrm {Lip}_b(X)$$ ψ ∈ Lip b ( X ) , and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $$Y\subset X$$ Y ⊂ X . In the latter case, the distribution-valued Ricci bound will be given by the signed measure $$\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}$$ κ = k m Y + ℓ σ ∂ Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and $$\ell $$ ℓ denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

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